@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-WG63ZQK3-T
  skos:related psr:-C5M4Q32X-B, psr:-B34655S6-R, psr:-B6WDN5KC-8, psr:-MRFZQ86M-2 ;
  skos:inScheme psr: ;
  skos:exactMatch <https://en.wikipedia.org/wiki/P-adic_number>, <https://fr.wikipedia.org/wiki/Nombre_p-adique> ;
  skos:definition """In number theory, given a prime number <span class="texhtml mvar" style="font-style:italic;">p</span>, the <b><span class="texhtml mvar" style="font-style:italic;">p</span>-adic numbers</b> form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; <span class="texhtml mvar" style="font-style:italic;">p</span>-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number <span class="texhtml mvar" style="font-style:italic;">p</span> rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number <span class="texhtml mvar" style="font-style:italic;">p</span>, a <span class="texhtml mvar" style="font-style:italic;">p</span>-adic number can be defined as a series
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle s=\\\\sum _{i=k}^{\\\\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\\\\cdots }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>s</mi>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>+</mo>
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>+</mo>
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle s=\\\\sum _{i=k}^{\\\\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\\\\cdots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ad992b963bfa46cf597d00daf64ba798a83fd0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:48.405ex; height:6.843ex;" alt="{\\\\displaystyle s=\\\\sum _{i=k}^{\\\\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\\\\cdots }"></span></dd></dl>
<br/>where <span class="texhtml mvar" style="font-style:italic;">k</span> is an integer (possibly negative), and each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle a_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="a_{i}"></span> is a integer such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle 0\\\\leq a_{i}<p.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>0</mn>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>&lt;</mo>
<br/>        <mi>p</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 0\\\\leq a_{i}&lt;p.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f17182b9eaa86a5121a307f7784b01140ea0f0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:11.205ex; height:2.509ex;" alt="{\\\\displaystyle 0\\\\leq a_{i}<p.}"></span> A <b><span class="texhtml mvar" style="font-style:italic;">p</span>-adic integer</b> is a <span class="texhtml mvar" style="font-style:italic;">p</span>-adic number such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k\\\\geq 0.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>k</mi>
<br/>        <mo>≥<!-- ≥ --></mo>
<br/>        <mn>0.</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle k\\\\geq 0.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a21e868e400e73512aa47ff205ebe6c2421919" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:6.119ex; height:2.343ex;" alt="{\\\\displaystyle k\\\\geq 0.}"></span>
<br/>In general the series that represents a <span class="texhtml mvar" style="font-style:italic;">p</span>-adic number is not convergent in the usual sense, but it is convergent for the <span class="texhtml mvar" style="font-style:italic;">p</span>-adic absolute value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle |s|_{p}=p^{-k},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo stretchy="false">|</mo>
<br/>        </mrow>
<br/>        <mi>s</mi>
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">|</mo>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>p</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>p</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle |s|_{p}=p^{-k},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46fc684774c8b4f9b7129085d64ec1074c16ba1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:10.725ex; height:3.509ex;" alt="{\\\\displaystyle |s|_{p}=p^{-k},}"></span> where <span class="texhtml mvar" style="font-style:italic;">k</span> is the least integer <span class="texhtml mvar" style="font-style:italic;">i</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle a_{i}\\
eq 0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>≠<!-- ≠ --></mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a_{i}\\
eq 0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66efc850e79e220d4f51bc2a2333af3d7325180" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.29ex; height:2.676ex;" alt="{\\\\displaystyle a_{i}\\
eq 0}"></span> (if all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle a_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="a_{i}"></span> are zero, one has the zero <span class="texhtml mvar" style="font-style:italic;">p</span>-adic number, which has <span class="texhtml">0</span> as its <span class="texhtml mvar" style="font-style:italic;">p</span>-adic absolute value). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">https://en.wikipedia.org/wiki/P-adic_number</a>)"""@en, """En mathématiques, et plus particulièrement en théorie des nombres, pour un nombre premier <span class="texhtml"><i>p</i></span> fixé, les <b>nombres <span class="texhtml"><i>p</i></span>-adiques</b> forment une extension particulière du corps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {Q} }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">Q</mi>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {Q} }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="\\\\mathbb{Q} "></span> des nombres rationnels, découverte par Kurt Hensel en 1897. Le corps commutatif <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {Q} _{p}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msub>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">Q</mi>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
         <mi>p</mi>
         </mrow>
         </msub>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {Q} _{p}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="\\\\mathbb{Q} _{p}"></span> des nombres <span class="texhtml"><i>p</i></span>-adiques peut être construit par complétion de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {Q} }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">Q</mi>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {Q} }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="\\\\mathbb{Q} "></span>, d'une façon analogue à la construction des nombres réels par les suites de Cauchy, mais pour une valeur absolue moins familière, nommée valeur absolue <span class="texhtml"><i>p</i></span>-adique.
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_p-adique">https://fr.wikipedia.org/wiki/Nombre_p-adique</a>)"""@fr ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:broader psr:-P93ST75Z-8 ;
  skos:prefLabel "p-adic number"@en, "nombre p-adique"@fr ;
  dc:modified "2023-08-04"^^xsd:date ;
  a skos:Concept ;
  skos:narrower psr:-L2ZDBS9T-G .

psr:-P93ST75Z-8
  skos:prefLabel "théorie des nombres"@fr, "number theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-WG63ZQK3-T .

psr:-B34655S6-R
  skos:prefLabel "polygone de Newton"@fr, "Newton polygon"@en ;
  a skos:Concept ;
  skos:related psr:-WG63ZQK3-T .

psr:-B6WDN5KC-8
  skos:prefLabel "solenoid"@en, "solénoïde"@fr ;
  a skos:Concept ;
  skos:related psr:-WG63ZQK3-T .

psr: a skos:ConceptScheme .
psr:-L2ZDBS9T-G
  skos:prefLabel "automorphic number"@en, "nombre automorphe"@fr ;
  a skos:Concept ;
  skos:broader psr:-WG63ZQK3-T .

psr:-C5M4Q32X-B
  skos:prefLabel "p-adic L-function"@en, "fonction L p-adique"@fr ;
  a skos:Concept ;
  skos:related psr:-WG63ZQK3-T .

psr:-MRFZQ86M-2
  skos:prefLabel "Hilbert symbol"@en, "symbole de Hilbert"@fr ;
  a skos:Concept ;
  skos:related psr:-WG63ZQK3-T .